X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=2a4586a01c9b36c546e3aeacbb599a14c3e581d8;hp=b97a09f3ff51f7655a30f486060afcd833ea3a50;hb=67ed83b1ad44c7590cf7d7c1ec3a079bc5140a61;hpb=112047cbf7e9935d097eebd7b3e5ad6dc395c0d8 diff --git a/week6.mdwn b/week6.mdwn index b97a09f3..2a4586a0 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -1,16 +1,45 @@ [[!toc]] -Types, OCAML ------------- +Polymorphic Types and System F +------------------------------ -OCAML has type inference: the system can often infer what the type of +[Notes still to be added. Hope you paid attention during seminar.] + + + + +Types in OCaml +-------------- + +OCaml has type inference: the system can often infer what the type of an expression must be, based on the type of other known expressions. -For instance, if we type +For instance, if we type # let f x = x + 3;; -The system replies with +The system replies with val f : int -> int = @@ -32,7 +61,7 @@ element: # (3) = 3;; - : bool = true -though OCAML, like many systems, refuses to try to prove whether two +though OCaml, like many systems, refuses to try to prove whether two functional objects may be identical: # (f) = f;; @@ -40,13 +69,27 @@ functional objects may be identical: Oh well. +[Note: There is a limited way you can compare functions, using the +`==` operator instead of the `=` operator. Later when we discuss mutation, +we'll discuss the difference between these two equality operations. +Scheme has a similar pair, which they name `eq?` and `equal?`. In Python, +these are `is` and `==` respectively. It's unfortunate that OCaml uses `==` for the opposite operation that Python and many other languages use it for. In any case, OCaml will accept `(f) == f` even though it doesn't accept +`(f) = f`. However, don't expect it to figure out in general when two functions +are equivalent. (That question is not Turing computable.) + + # (f) == (fun x -> x + 3);; + - : bool = false + +Here OCaml says (correctly) that the two functions don't stand in the `==` relation, which basically means they're not represented in the same chunk of memory. However as the programmer can see, the functions are extensionally equivalent. The meaning of `==` is rather weird.] -Booleans in OCAML, and simple pattern matching + + +Booleans in OCaml, and simple pattern matching ---------------------------------------------- Where we would write `true 1 2` in our pure lambda calculus and expect -it to evaluate to `1`, in OCAML boolean types are not functions -(equivalently, are functions that take zero arguments). Selection is +it to evaluate to `1`, in OCaml boolean types are not functions +(equivalently, they're functions that take zero arguments). Instead, selection is accomplished as follows: # if true then 1 else 2;; @@ -65,7 +108,7 @@ That is, # match true with true -> 1 | false -> 2;; - : int = 1 -Compare with +Compare with # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;; - : int = 9 @@ -73,7 +116,7 @@ Compare with Unit and thunks --------------- -All functions in OCAML take exactly one argument. Even this one: +All functions in OCaml take exactly one argument. Even this one: # let f x y = x + y;; # f 2 3;; @@ -87,7 +130,7 @@ Here's how to tell that `f` has been curry'd: After we've given our `f` one argument, it returns a function that is still waiting for another argument. -There is a special type in OCAML called `unit`. There is exactly one +There is a special type in OCaml called `unit`. There is exactly one object in this type, written `()`. So # ();; @@ -110,25 +153,45 @@ correct type is the unit: # f ();; - : int = 3 -Let's have some fn: think of `rec` as our `Y` combinator. Then +Now why would that be useful? + +Let's have some fun: think of `rec` as our `Y` combinator. Then - # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));; + # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));; val f : int -> int = # f 5;; - : int = 120 We can't define a function that is exactly analogous to our ω. -We could try `let rec omega x = x x;;` what happens? However, we can -do this: +We could try `let rec omega x = x x;;` what happens? + +[Note: if you want to learn more OCaml, you might come back here someday and try: + + # let id x = x;; + val id : 'a -> 'a = + # let unwrap (`Wrap a) = a;; + val unwrap : [< `Wrap of 'a ] -> 'a = + # let omega ((`Wrap x) as y) = x y;; + val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b = + # unwrap (omega (`Wrap id)) == id;; + - : bool = true + # unwrap (omega (`Wrap omega));; + + +But we won't try to explain this now.] - # let rec omega x = omega x;; + +Even if we can't (easily) express omega in OCaml, we can do this: + + # let rec blackhole x = blackhole x;; By the way, what's the type of this function? -If you then apply this omega to an argument, - # omega 3;; +If you then apply this `blackhole` function to an argument, + + # blackhole 3;; -the interpreter goes into an infinite loop, and you have to control-C +the interpreter goes into an infinite loop, and you have to type control-c to break the loop. Oh, one more thing: lambda expressions look like this: @@ -140,175 +203,101 @@ Oh, one more thing: lambda expressions look like this: (But `(fun x -> x x)` still won't work.) -So we can try our usual tricks: +You may also see this: + + # (function x -> x);; + - : 'a -> 'a = + +This works the same as `fun` in simple cases like this, and slightly differently in more complex cases. If you learn more OCaml, you'll read about the difference between them. + +We can try our usual tricks: - # (fun x -> true) omega;; + # (fun x -> true) blackhole;; - : bool = true -OCAML declined to try to evaluate the argument before applying the -functor. But remember that `omega` is a function too, so we can +OCaml declined to try to fully reduce the argument before applying the +lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language? + +Remember that `blackhole` is a function too, so we can reverse the order of the arguments: - # omega (fun x -> true);; + # blackhole (fun x -> true);; Infinite loop. Now consider the following variations in behavior: - # let test = omega omega;; - [Infinite loop, need to control c out] + # let test = blackhole blackhole;; + - # let test () = omega omega;; + # let test () = blackhole blackhole;; val test : unit -> 'a = # test;; - : unit -> 'a = # test ();; - [Infinite loop, need to control c out] + -We can use functions that take arguments of type unit to control -execution. In Scheme parlance, functions on the unit type are called +We can use functions that take arguments of type `unit` to control +execution. In Scheme parlance, functions on the `unit` type are called *thunks* (which I've always assumed was a blend of "think" and "chunk"). +Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like: + + let f = fun () -> blackhole () + in true + +terminate? + +Bottom type, divergence +----------------------- + +Expressions that don't terminate all belong to the **bottom type**. This is a subtype of every other type. That is, anything of bottom type belongs to every other type as well. More advanced type systems have more examples of subtyping: for example, they might make `int` a subtype of `real`. But the core type system of OCaml doesn't have any general subtyping relations. (Neither does System F.) Just this one: that expressions of the bottom type also belong to every other type. It's as if every type definition in OCaml, even the built in ones, had an implicit extra clause: + + type 'a option = None | Some of 'a;; + type 'a option = None | Some of 'a | bottom;; + +Here are some exercises that may help better understand this. Figure out what is the type of each of the following: + + fun x y -> y;; + + fun x (y:int) -> y;; + + fun x y : int -> y;; + + let rec blackhole x = blackhole x in blackhole;; + + let rec blackhole x = blackhole x in blackhole 1;; + + let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;; + + let rec blackhole x = blackhole x in (blackhole 1) + 2;; + + let rec blackhole x = blackhole x in (blackhole 1) || false;; + + let rec blackhole x = blackhole x in 2 :: (blackhole 1);; + +By the way, what's the type of this: + + let rec blackhole (x:'a) : 'a = blackhole x in blackhole + + +Back to thunks: the reason you'd want to control evaluation with thunks is to +manipulate when "effects" happen. In a strongly normalizing system, like the +simply-typed lambda calculus or System F, there are no "effects." In Scheme and +OCaml, on the other hand, we can write programs that have effects. One sort of +effect is printing (think of the [[damn]] example at the start of term). +Another sort of effect is mutation, which we'll be looking at soon. +Continuations are yet another sort of effect. None of these are yet on the +table though. The only sort of effect we've got so far is *divergence* or +non-termination. So the only thing thunks are useful for yet is controlling +whether an expression that would diverge if we tried to fully evaluate it does +diverge. As we consider richer languages, thunks will become more useful. + + Towards Monads -------------- -So the integer division operation presupposes that its second argument -(the divisor) is not zero, upon pain of presupposition failure. -Here's what my OCAML interpreter says: - - # 12/0;; - Exception: Division_by_zero. - -So we want to explicitly allow for the possibility that -division will return something other than a number. -We'll use OCAML's option type, which works like this: - - # type 'a option = None | Some of 'a;; - # None;; - - : 'a option = None - # Some 3;; - - : int option = Some 3 - -So if a division is normal, we return some number, but if the divisor is -zero, we return None: - -
```-let div (x:int) (y:int) =
-  match y with 0 -> None |
-               _ -> Some (x / y);;
-
-(*
-val div : int -> int -> int option = fun
-# div 12 3;;
-- : int option = Some 4
-# div 12 0;;
-- : int option = None
-# div (div 12 3) 2;;
-Characters 4-14:
-  div (div 12 3) 2;;
-      ^^^^^^^^^^
-Error: This expression has type int option
-       but an expression was expected of type int
-*)
-```
- -This starts off well: dividing 12 by 3, no problem; dividing 12 by 0, -just the behavior we were hoping for. But we want to be able to use -the output of the safe-division function as input for further division -operations. So we have to jack up the types of the inputs: - -
```-let div (x:int option) (y:int option) =
-  match y with None -> None |
-               Some 0 -> None |
-               Some n -> (match x with None -> None |
-                                       Some m -> Some (m / n));;
-
-(*
-val div : int option -> int option -> int option =
-# div (Some 12) (Some 4);;
-- : int option = Some 3
-# div (Some 12) (Some 0);;
-- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-```
- -Beautiful, just what we need: now we can try to divide by anything we -want, without fear that we're going to trigger any system errors. - -I prefer to line up the `match` alternatives by using OCAML's -built-in tuple type: - -
```-let div (x:int option) (y:int option) =
-  match (x, y) with (None, _) -> None |
-                    (_, None) -> None |
-                    (_, Some 0) -> None |
-                    (Some m, Some n) -> Some (m / n);;
-```
- -So far so good. But what if we want to combine division with -other arithmetic operations? We need to make those other operations -aware of the possibility that one of their arguments will trigger a -presupposition failure: - -
```-let add (x:int option) (y:int option) =
-  match (x, y) with (None, _) -> None |
-                    (_, None) -> None |
-                    (Some m, Some n) -> Some (m + n);;
-
-(*
-val add : int option -> int option -> int option =
-# add (Some 12) (Some 4);;
-- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-```
- -This works, but is somewhat disappointing: the `add` operation -doesn't trigger any presupposition of its own, so it is a shame that -it needs to be adjusted because someone else might make trouble. - -But we can automate the adjustment. The standard way in OCAML, -Haskell, etc., is to define a `bind` operator (the name `bind` is not -well chosen to resonate with linguists, but what can you do): - -
```-let bind (x: int option) (f: int -> (int option)) =
-  match x with None -> None |
-               Some n -> f n;;
-
-let add (x: int option) (y: int option)  =
-  bind x (fun x -> bind y (fun y -> Some (x + y)));;
-
-let div (x: int option) (y: int option) =
-  bind x (fun x -> bind y (fun y -> if (0 = y) then None else Some (x / y)));;
-
-(*
-#  div (div (Some 12) (Some 2)) (Some 4);;
-- : int option = Some 1
-#  div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-```
- -Compare the new definitions of `add` and `div` closely: the definition -for `add` shows what it looks like to equip an ordinary operation to -survive in dangerous presupposition-filled world. Note that the new -definition of `add` does not need to test whether its arguments are -None objects or real numbers---those details are hidden inside of the -`bind` function. - -The definition of `div` shows exactly what extra needs to be said in -order to trigger the no-division-by-zero presupposition. - -For linguists: this is a complete theory of a particularly simply form -of presupposition projection (every predicate is a hole). +This has now been moved to the start of [[week7]]. +